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Applicable AnalysisAn International Journal

ISSN: 0003-6811 (Print) 1563-504X (Online) Journal homepage: http://www.tandfonline.com/loi/gapa20

A stabilized finite element method for theconvection dominated diffusion optimal controlproblem

Zhifeng Weng, Jerry Zhijian Yang & Xiliang Lu

To cite this article: Zhifeng Weng, Jerry Zhijian Yang & Xiliang Lu (2015): A stabilized finiteelement method for the convection dominated diffusion optimal control problem, ApplicableAnalysis, DOI: 10.1080/00036811.2015.1114606

To link to this article: http://dx.doi.org/10.1080/00036811.2015.1114606

Published online: 26 Nov 2015.

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APPLICABLE ANALYSIS, 2015http://dx.doi.org/10.1080/00036811.2015.1114606

A stabilized finite element method for the convection dominateddiffusion optimal control problem

Zhifeng Wenga,b, Jerry Zhijian Yanga,c and Xiliang Lua,c

aSchool of Mathematics and Statistics, Wuhan University, Wuhan, P.R. China; bSchool of Mathematics Science,Huaqiao University, Quanzhou, P.R. China; cHubei Key Laboratory of Computational Science, Wuhan University,Wuhan, P.R. China

ABSTRACTIn this paper, a stabilized finite element method for optimal controlproblems governed by a convection dominated diffusion equation isinvestigated. The state and the adjoint variables are approximated bypiecewise linear continuous functions with bubble functions. The controlvariable either is approximated by piecewise linear functions (called thestandard method) or is not discretized directly (called the variationaldiscretization method). The stabilization term only depends on bubblefunctions, and the projection operator can be replaced by the differenceof two local Gauss integrations. A priori error estimates for both methodsare given and numerical examples are presented to illustrate the theoreticalresults.

ARTICLE HISTORYReceived 4 May 2015Accepted 26 October 2015

COMMUNICATED BYC. Bacuta

KEYWORDSConvection dominateddiffusion equation; stabilizedfinite element method;optimal control problem;variational discretization

AMS SUBJECTCLASSIFICATIONS65M60; 76D07; 65M12

1. Introduction

The convection dominated diffusion equations arise in many engineering applications. When thediffusion coefficient is very small, the convection dominated diffusion equations have a multiscalebehavior between the diffusion and the advection, which brings enormous challenges in the endeavorof numerical approximation.

The standard Galerkin finite element methods for the convection dominated diffusion problemsmay produce approximate solution with large nonphysical oscillations unless the mseh size isvery small which depends on the diffusion coefficient. A lot of methods have been developed toovercome this problem, for example, the Galerkin least square method,[1,2] streamline upwindPetrov Galerkin (SUPG) method,[3,4] the residual-free bubbles method,[5,6] the local projectionstabilization.[7–9] Burman et al. [10] proposed an edge-stabilizationGalerkinmethod to approximatethe convection dominated diffusion equations. Knobloch [11] introduces a generalization of thelocal projection stabilization for the convection diffusion reaction equations which allows us touse local projection spaces defined on overlapping sets. Wu et al. [12] used the streamline upwindPetrov–Galerkin to solve the convection dominated problems in order to eliminate overshoots andundershoots produced by the convection term in the meshless local Petrov–Galerkin method. Songet. al. [13] presented variational multiscale method based on bubble functions for convection domi-nated diffusion equation.

However, for the optimal control problem governed by convection dominated diffusion equations,some stabilization techniques are not straightforward to apply. One reason is that the two com-mon approaches for PDE-constraint optimization problem, i.e. “first-optimize-then-discretize” and

CONTACT Xiliang Lu xllv.math@whu.edu.cn© 2015 Taylor & Francis

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2 Z. WENG ET AL.

“first-discretize-then-optimize,” are not equivalent if the stabilized term is not properly chosen.For example, the streamline upwind Galerkin method (SUPG) was not well suited for the dualitytechniques applied in optimal control when one considers the method for discretizing the stateand the adjoint equation in the optimality system. Becker et al. [14] considered the stabilizationfinite element discretization based on local projections for the convection diffusion equation, wherethe symmetrical penalty terms were applied. Then finding the optimality system to the optimalcontrol problem on the continuous level and then discretizing the optimality system appropriately isequivalent to considering the optimal control problem on the discrete level with stabilization term.

Recently, Yan et al. [15] studied a priori and a posteriori error estimates of edge stabilizationGalerkin method for the optimal control problem governed by convection dominated diffusionequation. Zhou et al. [16] presented the local discontinuous Galerkin method for optimal controlproblemgovernedby convectiondiffusion equations. Yücel et al. [17] investigated distributed optimalcontrol of convection diffusion reaction equations using discontinuous Galerkin methods. Akmanet al. [18] developed a priori error analysis of the upwind symmetric interior penalty Galerkinmethodfor the optimal control problems governed by unsteady convection diffusion equations.

Influenced by theworkmentioned above,wewill apply the bubble stabilizationGalerkinmethodofSong et al. [13] to the discretization of optimal control problems governed by convection dominateddiffusion equations. Firstly, we obtain the continuous optimality system, which contains the stateequation, the adjoint equation, and the optimality condition, which is given in terms of a variationalinequality. Then we give the discrete optimal control problem by using the bubble stabilizationGalerkin method to approximate the state equation, whose optimality system then coincides withthat obtained by discretizing the state and adjoint state in the continuous optimality system by finiteelements with bubble stabilization, and the control is approximated by piecewise linear functions.

However, the above approach cannot recover the full accuracy for the control variable. To obtaina better approximation to the control variable, Hinze introduced a variational discretization conceptfor optimal control problems and derived an optimal a priori error estimate for the control in [19,20].The variational discretization concept for optimal control problems with control constraints utilizesthe first-order optimality conditions and the discretization of the state and adjoint equations, thenthe control variable is obtained by the projection of the adjoint state. In this paper, we will combinevariational discretization and the stabilized finite element method for the designed control problems.The half-order higher approximation for the control variable can be obtained.

The remainders of this paper is organized as follows. In Section 2, we present the model problemand derive the optimality system. The bubble stabilization Galerkinmethod based on two local Gaussintegrations for the optimal control problem is given in Section 3. In Section 4, a priori error estimatefor the discrete control variables is derived and In Section 5, we prove the a priori error estimateusing variational discretization. Then in Section 6, numerical experiments are shown to verify thetheoretical results. Finally, in Section 7, we conclude with a summary and possible extensions.

The standard Sobolev spaceWm,p(�) is equipped with a norm ‖ · ‖m,p. For p = 2, let Hm(�) =Wm,2(�) and write ‖ · ‖m = ‖ · ‖m,2 for m ≥ 0. (·, ·) is the inner product in L2(�) or [L2(�)]2. Weuse c or C to denote a generic positive constant whose value may change from place to place, butremains independent of the mesh size h and the diffusion coefficient ε.

2. Model problem

Consider the following optimal control problem governed by convection dominated diffusionequations:

minu∈KJ(y, u) = 12‖y − yd‖20 + α

2‖u‖20. (1)

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APPLICABLE ANALYSIS 3

subject to:

β · ∇y − ε�y + sy = f + Bu, in �, (2)y = 0, on ∂�, (3)

where � ⊂ R2 is a bounded domain with Lipschitz boundary ∂�, α > 0 is a positive constant, β is

either a constant vector or divergence free velocity field, ε is small diffusion coefficient, f ∈ L2(�)

and s ≥ s0 > 0 is the reaction coefficient. The target state yd ∈ L2(�), and the control variable uis supported in a sub-domain ω ⊂ �. The operator B : L2(ω) �→ L2(�) is the extension-by-zerooperator, hence its adjoint operator B∗ : L2(�) �→ L2(ω) is the restriction operator and B∗B isidentity operator in L2(ω). The admissible set K ⊂ L2(ω) is given by

K = {u ∈ L2(ω); c ≤ u ≤ c, a.e. in ω},

where c < c are two constants for the box constraints.Define a continuous bilinear forms a(·, ·) on H1

0 (�) × H10 (�) by

a(y, v) = ε(∇y,∇v) + (β · ∇y, v) + (sy, v),

then the weak formulation of the state Equations (2)–(3) is: to find y ∈ H10 (�) such that

a(y, v) = (f + Bu, v), ∀v ∈ H10 (�).

To find the optimality system, one needs the adjoint equation

−β · ∇p − ε�p + sp = y − yd , in �,p = 0, on ∂�,

and its weak formulation reads: to find p ∈ H10 (�) such that

b(p,w) = (y − yd ,w), ∀w ∈ H10 (�),

where

b(p,w) = ε(∇p,∇w) − (β · ∇p,w) + (sp,w).

Denote by A : K �→ H10 (�) the solution operator of the state Equations (2) and (3) and introduce

the reduced cost functional j : K �→ R by :

j(u) = J(Au, u),

then we can eliminate the state equation and to reformulate the optimization problem as:

Minimize j(u), ∀u ∈ K .

The reduced cost functional j is quadratic, its first- and second-order derivatives satisfies: (see [14,21]):

j′(u)(δu) = (B∗p + αu, δu), (4)

j′′(u)(δu, δu) ≥ α‖δu‖20, ∀δu ∈ K .

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4 Z. WENG ET AL.

Since K is closed and convex, there exists a unique solution to optimal control problem (1)–(3),and there exists an adjoint function (i.e. Lagrange multiplier) p ∈ H1

0 (�), such that (y, p, u) satisfiesthe following optimality system (see e.g. [14,22]:

a(y, v) = (f + Bu, v), ∀v ∈ H10 , (5)

b(p,w) = (y − yd ,w), ∀w ∈ H10 , (6)

(αu + B∗p, u − u) ≥ 0, ∀u ∈ K . (7)

For the boxed constraints, the variational inequality (7) can be represented by the pointwiseprojection:

u = Proj[c,c](−B∗p

α

),

where the projection operator is defined by

Proj[c,c] = max (c, min (c, u)).

When diffusion coefficient ε is very small, the P1 nodal element discretization for the state Equation(5) is unstable. In order to improve the computational stability and accuracy, one should adopt thestabilized methods. In this work, we will use the bubble stabilization Galerkin scheme based on twolocal Gauss integrations (see e.g. [23]) to deal with the state equation and the adjoint equation in theoptimality system (5)–(7).

3. Bubble stabilized finite element methods based on two local Gauss integrations

Let Th be the quasi-uniform triangulations of the domain � with the mesh size h = max{diam(T) :T ∈ Th} as in [24]. The finite element spaces are defined by

Mh = {yh ∈ C0(�)|yh|T ∈ P1(T), ∀T ∈ Th},Mb

h = {vh ∈ C0(�)|vh|T ∈ P1(T) ⊕ B(T), ∀T ∈ Th},

where P1(T) is the space of first-order polynomials on T and B(T) denotes the space of bubblefunctions. The bubble function defined as follows:

B(T) = {vh ∈ C(T)|vh ∈ Span{λ0λ1λ2}}, ∀T ∈ Th,

where λi are area coordinates on T , i = 0, 1, 2. The area coordinate is also known as a trianglebarycentre coordinate, where the three components (λ0, λ1, λ2) are of the ratio between the area ofthe three triangles and the area of themother triangle.Wewill also need the piecewise constant vectorspace

R0 = {Vh ∈ (L2(�))2|Vh|T ∈ (P0(T))2, ∀T ∈ Th},where P0(T) is piecewise constant on T .

We introduce a stabilized term G(·, ·) that defines the inner product inMbh as:

G(yh, vh) = ν((∇yh,∇vh) − (�∇yh,�∇vh)), (8)

where ν is a nonnegative function depending on the mesh size h, and � : (L2(�))2 �→ R0 is the L2orthogonal projection. This stabilization formulation can be casted in the framework of variationalmultiscale method (see [13]). From the definition of �, one can obtain

(∇yh − �∇yh,�∇vh) = 0. (9)

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APPLICABLE ANALYSIS 5

An attractive feature of this stabilization approach is the flexibility in that the stabilized operator �

can be represented by the difference of local Gauss integration (see e.g. [13]):

G(yh, vh) = ν∑T∈Th

⎛⎜⎝

∫T ,k

∇yh∇vhd� −∫T ,1

∇yh∇vh d�

⎞⎟⎠ ∀yh, vh ∈ Mb

h ,

where∫T ,i g(x) indicates an approximation Gauss integral over T which is exact for polynomials of

degree i, i = 1, k, k ≥ 2.To prove convergence of stabilized solutions, we first notice that [24]

‖�v‖ ≤ C‖v‖, ∀v ∈ (L2(�))2, (10)‖v − �v‖ ≤ Ch‖v‖1, ∀v ∈ (H1(�))2. (11)

Using the stabilization finite element method as above, a discretization of the optimal controlproblem (1)–(3) can be defined as follows:

minuh∈Kh

J(yh, uh). (12)

subject to:

a(yh, vh) + G(yh, vh) = (f + Buh, vh), ∀vh ∈ Mbh , (13)

whereKh is the discrete control space. For the standardmethod,Kh = Mh∩K , and for the variationaldiscretization method Kh = K . Next we give a discrete solution operator Ah : K → Mb

h defined by:

a(Ahu, vh) + G(Ahu, vh) = (f + Bu, vh), ∀vh ∈ Mbh . (14)

and the discrete reduced cost functional

jh(u) = J(Ahu, u).

Similar as the continuous case, by introducing thediscrete adjoint variableph (which satisfiesEquation(18)), we can find the first- and second-order derivative of jh:

j′h(u)(δu) = (B∗ph + αu, δu), (15)j′′h(u)(δu, δu) ≥ α‖δu‖20. ∀δu ∈ K . (16)

Moreover, since Kh is also convex and closed, there exists a unique solution (yh, uh) to (12)–(13) anda corresponding adjoint variable (Lagrange multiplier) ph ∈ Mb

h , such that (yh, ph, uh) satisfies theoptimality system:

a(yh, vh) + G(yh, vh) = (f + Buh, vh), ∀vh ∈ Mbh , (17)

b(ph,wh) + G(ph,wh) = (yh − yd ,wh), ∀wh ∈ Mbh , (18)

(αuh + B∗ph, uh − uh) ≥ 0, ∀uh ∈ Kh, (19)

Remark 3.1: In our setting the approaches “discretize-then-optimize” and “optimize-then-discretize” coincide due to the fact that G(yh, vh) is a symmetric bilinear form.

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6 Z. WENG ET AL.

Recall that by standard approximation theory, for a regular family of elements, there exists aninterpolation Ph : H1

0 (�) ∩ H2(�) �→ Mbh with the following properties in [24]

‖Phv‖0 ≤ c‖v‖0, (20)‖v − Phv‖0 + h‖(v − Phv)‖1 ≤ ch2‖v‖2. (21)

Next, we will give the estimate of the term G(Phv,Phv).Lemma 3.1: For any v ∈ H1

0 (�) ∩ H2(�) and the following estimate holds

G(Phv,Phv) ≤ Cνh2‖v‖22. (22)

Proof: From the definition of the term G(Phv,Phv), We can obtain

G(Phv,Phv) = G(v + Phv − v, v + Phv − v) ≤ 2(G(v, v) + G(v − Phv, v − Phv)).

For the first term, we can get from the interpolation property of (11)

G(v, v) = ν‖∇v − �∇v‖20 ≤ Cνh2‖v‖22.

For the second term, we can obtain by the stability of (10)

G(v − Phv, v − Phv) ≤ Cν‖∇v − Ph∇v‖20 ≤ Cνh2‖v‖22.

The proof has been completed. �

4. A priori error analysis for the standardmethod

In this section, we consider a priori error estimates for the optimal control problem (5)–(7) andits bubble stabilization Galerkin approximation (17)–(19) for the standard method. For the sake ofsimplicity, we choose ω = � and hence B = I is an identity operator. To prove the a priori errorestimate, we first introduce a norm:

‖|v‖|2 = ‖s1/20 v‖20 + ‖ε1/2∇v‖20 + G(v, v).

In the following lemma, we give an error estimate for the discretization of the state equation with anadditional perturbation in the right hand side.Lemma 4.1: Let for u ∈ K , y = Au ∈ H1

0 (�) ∩ H2(�) be the associated solution of the stateEquations (5) and (7), and for z ∈ K , yh = Ahz ∈ Mb

h be the associated discrete solution, i.e:

a(yh, vh) + G(yh, vh) = (f + z, vh), ∀vh ∈ Mbh . (23)

Then the following estimate holds:

‖|y − yh‖| ≤ ‖u − z‖0 + cτh‖y‖2. (24)

whereτ = ε1/2 + h + ν1/2 + ε−1/2h. (25)

Proof: From (5) and (23), we can have

a(y − yh, vh) = G(yh, vh) + (u − z, vh), ∀vh ∈ Mbh . (26)

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APPLICABLE ANALYSIS 7

Let y − yh = δ + η withδ = y − Phy, η = Phy − yh.

By (21), we get:‖δ‖0 ≤ ch2‖y‖2. (27)

and‖|δ‖| ≤ ch2‖y‖2 + cε1/2h‖y‖2 + cν1/2h‖y‖2. (28)

For η ∈ Mbh and from (26), we can obtain

‖|η‖|2 = a(η, η) + G(η, η) = (u − z, η) + G(Phy, η) − a(δ, η).

For the first term, applying Cauchy–Schwarz inequality, we get:

(u − z, η) ≤ C‖u − z‖0‖η‖0 ≤ ‖u − z‖0‖|η‖|.

For the second term, by Cauchy–Schwarz inequality and (22), we can have

G(Phy, η) ≤ (G(Phy,Phy))1/2(G(η, η))1/2 ≤ cν1/2h‖y‖2‖|η‖|.

For the third term we derive:

a(δ, η) ≤ ‖|δ‖|‖|η‖| + |(β · ∇δ, η)|,

where|(β · ∇δ, η)| = |(δ,β · ∇η)| ≤ cε−1/2‖δ‖‖ε1/2∇η‖ ≤ cε−1/2‖|η‖|‖δ‖0.

Therefore we can obtain

‖|η‖| ≤ ‖u − z‖0 + cε−1/2‖δ‖0 + cν1/2h‖y‖2 + ‖|δ‖|.

Applying the triangle inequality and using (27) and (28), we can obtain (24). �Remark 4.1: The above estimation involves the diffusion coefficient ε. It is not easy to obtain theε-free estimation in the convection dominated case, see the discussion in [10,13] and the referencescited there. In our numerical tests, the mesh size h is independent of the diffusion coefficient ε. Thestabilization parameter ν is chosen as the scale of O(h) in order to stabilize the convective termappropriately.

By a similar argument as in Lemma 4.1, we derive the error estimate for the adjoint equation.Lemma 4.2: Let for u ∈ K , p ∈ H1

0 (�) ∩ H2(�) be the associated solution of the state Equation (5)and (7), and for z ∈ K , ph(z) ∈ Mb

h denote the associated adjoint discrete solution, i.e:

b(ph,wh) + G(ph,wh) = (yh − yd ,wh), ∀wh ∈ Mbh ,

then the following estimate holds:

‖|p − ph‖| ≤ ‖u − z‖0 + cτh(‖y‖2 + ‖p‖2). (29)

Remark 4.2: when u = z, The error estimates of bubble-stabilization Galerkin method forconvection-dominated diffusion equation are derived from (24) and (29)

‖|y − yh‖| ≤ cτh‖y‖2. (30)

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8 Z. WENG ET AL.

and‖|p − ph‖| ≤ cτh‖p‖2. (31)

In addition, we introduce the inactive set in optimum:

�I = x ∈ � : c < u(x) < c.

Note, that u = −p/α on �I and therefore u|�I ∈ H2(�I). Using this subdomain �I , we define anorm that will be needed later in the paper

‖u‖2,ad =(‖u‖2W1,∞(�)

+ ‖�u‖2L2(�I )

)1/2.

Then from [14], it has constructed a special interpolation uI ∈ Kh of the solution u ∈ W1,∞(�),which fulfills the following conditions:

j′(u)(r − uI) ≥ 0, ∀r ∈ K , (32)

‖u − uI‖0 ≤ cαh3/2‖u‖2,ad . (33)

Theorem 4.1: Let (y, p, u) and (yh, ph, uh) denote the solutions to (5)–(7) and (17)–(19), respectively.Assume that y, p ∈ H2(�). Then we have:

‖u − uh‖0 ≤ ch3/2‖u‖2,ad + cτh(‖y‖2 + ‖p‖2). (34)

and‖|y − yh‖| + ‖|p − ph‖| ≤ ch3/2‖u‖2,ad + cτh(‖y‖2 + ‖p‖2). (35)

where τ is defined as in (25).Proof: From (16), we can get

α‖uI − uh‖2 ≤ j′′h(uh)(uI − uh, uI − uh)= j′h(uI)(uI − uh) − j′h(uh)(uI − uh).

By (19) and (32) with r = uh. We obtain:

−j′h(uh)(uI − uh) ≤ 0 ≤ −j′(u)(uI − uh).

Hence,α‖uI − uh‖2 ≤ j′h(uI)(uI − uh) − j′(u)(uI − uh).

Using (4) and (15) and Cauchy–Schwarz inequality, we can obtain:

α‖uI − uh‖2 ≤ (ph − p, uI − uh) + α(uI − u, uI − uh)≤ ‖ph − p‖‖uI − uh‖ + α‖uI − u‖‖uI − uh‖. (36)

where p is the associated adjoint state to u and ph is the associated discrete adjoint state to uI . From(29), we have:

α‖uI − uh‖ ≤ ‖ph − p‖ + α‖uI − u‖≤ (1 + α)‖u − uI‖0 + cτh(‖y‖2 + ‖p‖2). (37)

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APPLICABLE ANALYSIS 9

Then, using (37) and (33) and applying the triangle inequality, we have

‖u − uh‖ ≤ 1 + α

α2 ch3/2‖u‖2,ad + cαhτ(‖y‖2 + ‖p‖2).

we have completed the proof of (34). Then let Ahu and ph(u) be the bubble stabilization Galerkinsolution of the following equation

a(Ahu, vh) + G(Ahu, vh) = (f + u, vh), ∀vh ∈ Mbh , (38)

b(ph(u),wh) + G(ph(u),wh) = (Ahu − yd ,wh), ∀wh ∈ Mbh . (39)

From (18) and (39), we can deduce that

b(ph − ph(u),wh) + G(ph − ph(u),wh) = (yh − Ahu,wh), ∀wh ∈ Mbh .

Let wh = ph − ph(u), we have that

‖|ph − ph(u)‖| ≤ C‖|yh − Ahu‖|. (40)

Similarly, using (17) and (38) and setting vh = yh − Ahu, it can be proved that

‖|yh − Ahu‖| ≤ C‖u − uh‖0. (41)

Using (30) and (41) and applying the triangle inequality, we can obtain

‖|y − yh‖| ≤ ‖|y − Ahu‖| + ‖|yh − Ahu‖|≤ Cτh‖y‖2 + C‖u − uh‖. (42)

Similarly, combining (31), (40), and (41), we derive

‖|p − ph‖| ≤ ‖|p − ph(u)‖| + ‖|ph − ph(u)‖|≤ Cτh‖p‖2 + C‖u − uh‖. (43)

Summing up, (34), (42) and (43) prove the theoretical result (35). �Remark 4.3: For the piecewise linear case, the optimal order of control variable isO(h2).However,its the convergence order is only O(h3/2) , which is caused by the fact that umay not be smooth nearthe free boundary even if y and p are smooth there.

5. A priori error estimates for the variational discretizationmethod

Next, we consider a priori error estimates for the optimal control problem (5)–(7) and its bubblefunction stabilization Galerkin approximation (17)–(19) for the variational discretization method,i.e. Kh = K .Theorem 5.1: Let (u, p, y) and (uh, ph, yh) denote the solutions to (5)–(7) and (17)–(19)withKh = K,respectively. Assume that y, p ∈ H2(�), then we have:

‖u − uh‖0 + ‖|y − yh‖| + ‖|p − ph‖| ≤ cτh(‖y‖2 + ‖p‖2), (44)

whereτ = ε1/2 + h + ν1/2 + ε−1/2h.

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10 Z. WENG ET AL.

Proof: Let u = u in (19) and u = uh in (7), then add the resulting inequalities. We can obtain

α‖u − uh‖20 ≤ (p − ph, uh − u)= (p − ph(u), uh − u) + (ph(u) − ph, uh − u) = I + II

whereI = (p − ph(u), uh − u), II = (ph(u) − ph, uh − u)

and ph(u) is the solution of the following equation:

b(ph(u), qh) + G(ph(u), qh) = (y(u) − yd , qh), ∀qh ∈ Mbh . (45)

By Cauchy–Schwarz inequality and Young’s inequality, we can obtain that

|I| = |(p − ph(u), uh − u)| (46)≤ ‖p − ph(u)‖0‖uh − u‖0≤ 1

2α‖p − ph(u)‖20 + α

2‖uh − u‖20.

Choosing vh = ph − ph(u) in (17) and (14) respectively, and subtracting (17) from (14) implies that

II = a(Ahu − yh, ph − ph(u)) + G(Ahu − yh, ph − ph(u)).

By definition and using (18) and (45) we have

II = a(Ahu − yh, ph − ph(u)) + G(Ahu − yh, ph − ph(u))= b(ph − ph(u),Ahu − yh) + G(ph − ph(u),Ahu − yh)= (y − yh, yh − Ahu)= (y − yh, yh − y) + (y − yh, y − Ahu)

≤ −12‖y − yh‖20 + 1

2‖y − Ahu‖20. (47)

Combining (46) and (47) it gives

α‖u − uh‖20 + ‖y − yh‖20 ≤ 1α

‖p − ph(u)‖20 + ‖y − Ahu‖20.

From above inequality and the a priori estimation for PDE, e.g. (30) and (31), we can get

‖u − uh‖0 ≤ c(‖p − ph(u)‖0 + ‖y − Ahu‖0) ≤ cτh(‖y‖2 + ‖p‖2).

Finally we need to estimate |||y − yh||| and |||p − ph|||. From the definition of Ahu in (18) and ph(u)in (45) and the stability of the Galerkin method we deduce that

|||yh − Ahu||| ≤ ‖u − uh‖0 (48)

and|||ph − ph(u)||| ≤ C‖y − yh‖0 ≤ C|||y − yh|||. (49)

Therefore the triangle inequality combined with (30), (31), (48) and (49) gives (44). �Remark 5.1: In comparison with Theorems 4.1, 5.1 gives the convergence order of control variableis O(h3/2) in theory. But for variational discretization, the estimate of control variable is O(h2) from

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APPLICABLE ANALYSIS 11

numerical experiments because the method is not to discretize the space of admissible controls but toimplicitly utilize the first-order optimality conditions and the relation between co-state and controlfor the discretization of the control. For more information on the features of u, please refer to [25].

6. Numerical experiments

In this section, we present two numerical examples to demonstrate our theoretical results. Thegoverning equation in the first and the second examples are the linear and the nonlinear stateequations, respectively. The state and the adjoint variable are approximated by Mb

h and the controlvariable is approximated by Kh. The stabilization parameter is given by ν = Ch and the choiceof C accords with the discussion in [26]. The discrete optimality system (17)–(19) is solved by thesemismooth Newton method, see e.g. [25,27,28].Example 6.1:

minu∈K12

∫�

(y − yd)2dx + α

2

∫�

u2dx, (50)

subject to:

β · ∇y − ε�y + y = f + u, in �, (51)y = 0, on ∂�. (52)

and the adjoint equation as:

−β · ∇p − ε�p + p = y − yd , in �,p = 0, on ∂�.

Case a. Consider problem (50)–(52) with β = (1,√2), α = 1 and ε = 10−8. Computational domain

� = [0, 1]×[0, 1] and the admissible set isK = {u ∈ L2(�),−3 ≤ u ≤ 6}. To validate the theoreticalresults we consider the following given solution:

y = sin (2πx1) sin (2πx2),p = −π2 sin (2πx1) sin (2πx2),u = min{−3,max{6,−p}},

and the corresponding f and yd are obtained by inserting y, p, u into the optimality system.In this case, the numerical solutions are computed on a series of triangular meshes, which are

created from consecutive global refinement of an initial coarse mesh. At each refinement, everytriangle is divided into four congruent triangles. Tables 1 and 2 shows the error of the given schemes.

Table 1. Error estimates with ν = 0.1 h for standard method.

1/h |||y − yh||| Rate |||p − ph||| Rate ‖u − uh‖0 Rate

20 3.77e−2 3.59e−1 1.01e−140 1.30e−2 1.53 1.24e−1 1.53 3.98e−2 1.3860 6.96e−3 1.55 6.72e−2 1.51 2.18e−2 1.4880 4.50e−3 1.52 4.35e−2 1.51 1.47e−2 1.37100 3.21e−3 1.52 3.11e−2 1.51 1.07e−2 1.43120 2.43e−3 1.53 2.36e−2 1.51 8.02e−3 1.59

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12 Z. WENG ET AL.

Table 2. Error estimates with ν = 0.1 h for variational discretization.

1/h |||y − yh||| Rate |||p − ph||| Rate ‖u − uh‖0 Rate

20 3.66e−2 3.59e−1 4.70e−240 1.26e−2 1.54 1.24e−1 1.53 1.17e−2 2.0160 6.83e−3 1.52 6.72e−2 1.51 5.16e−3 2.0180 4.42e−3 1.51 4.35e−2 1.51 2.89e−3 2.01100 3.12e−3 1.51 3.11e−2 1.51 1.85e−3 2.00120 2.40e−3 1.51 2.36e−2 1.51 1.28e−3 2.00

IsoValue-0.95-0.85-0.75-0.65-0.55-0.45-0.35-0.25-0.15-0.050.050.150.250.350.450.550.650.750.850.95

IsoValue-9.37612-8.38916-7.4022-6.41524-5.42828-4.44132-3.45436-2.4674-1.48044-0.493480.493481.480442.46743.454364.441325.428286.415247.40228.389169.37612

IsoValue-2.775-2.325-1.875-1.425-0.975-0.525-0.0750.3750.8251.2751.7252.1752.6253.0753.5253.9754.4254.8755.3255.775

IsoValue-0.95113-0.851015-0.7509-0.650784-0.550669-0.450553-0.350438-0.250323-0.150207-0.05009180.05002360.1501390.2502540.350370.4504850.5506010.6507160.7508310.8509470.951062

IsoValue-9.38944-8.40108-7.41272-6.42436-5.43601-4.44765-3.45929-2.47094-1.48258-0.4942230.4941351.482492.470853.459214.447565.435926.424287.412638.400999.38935

IsoValue-2.775-2.325-1.875-1.425-0.975-0.525-0.0750.3750.8251.2751.7252.1752.6253.0753.5253.9754.4254.8755.3255.775

IsoValue-0.95156-0.851388-0.751217-0.651046-0.550875-0.450703-0.350532-0.250361-0.150189-0.05001810.05015320.1503240.2504960.3506670.4508380.551010.6511810.7513520.8515240.951695

IsoValue-9.38923-8.40089-7.41255-6.42421-5.43587-4.44752-3.45918-2.47084-1.4825-0.4941560.4941861.482532.470873.459214.447555.43596.424247.412588.400929.38926

IsoValue-2.775-2.325-1.875-1.425-0.975-0.525-0.0750.3750.8251.2751.7252.1752.6253.0753.5253.9754.4254.8755.3255.775

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 1. Plots of exact solution y, p, u (top) and numerical solution yh , ph , uh (middle) for the discrete control variables andnumerical solution yh , ph , uh for variational discretization (bottom) with linear case.

One can find that two schemes obtain the similar order for the primal and the adjoint states, but thevariational discretization get half order higher than the standard method, see the discussion in [20].

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APPLICABLE ANALYSIS 13

(a)

4 5 6 7 8 9 10−8

−7

−6

−5

−4

−3

−2

−1

0

log2(h)

log2(error)

|||y−yh||||||p−ph|||||u−uh||

(b)

4 5 6 7 8 9 10−9

−8

−7

−6

−5

−4

−3

−2

−1

0

log2(h)

log2(error)

|||y−yh||||||p−ph|||||u−uh||

(c)

Figure 2. (a) L-shaped domain, (b) convergence orders of y, p, u in energy norm, and (c) convergence orders of y, p, u in energynorm for variational discretization.

IsoValue0.01886650.05659940.09433230.1320650.1697980.2075310.2452640.2829970.320730.3584630.3961960.4339290.4716610.5093940.5471270.584860.6225930.6603260.6980590.735792

IsoValue-0.182204-0.17286-0.163517-0.154173-0.144829-0.135485-0.126141-0.116798-0.107454-0.0981099-0.0887661-0.0794223-0.0700785-0.0607347-0.0513909-0.0420471-0.0327033-0.0233595-0.0140157-0.0046719

IsoValue0.5342190.6026570.6710950.7395330.8079710.8764090.9448471.013291.081721.150161.21861.287041.355481.423911.492351.560791.629231.697671.76611.83454

IsoValue0.01916140.05748410.09580680.134130.1724520.2107750.2490980.287420.3257430.3640660.4023890.4407110.4790340.5173570.5556790.5940020.6323250.6706480.708970.747293

IsoValue-0.174697-0.165738-0.156779-0.147821-0.138862-0.129903-0.120944-0.111985-0.103026-0.0940677-0.0851088-0.07615-0.0671912-0.0582324-0.0492735-0.0403147-0.0313559-0.0223971-0.0134382-0.00447941

IsoValue0.5322940.5968820.6614710.7260590.7906470.8552350.9198240.9844121.0491.113591.178181.242761.307351.371941.436531.501121.565711.630291.694881.75947

(a) (b) (c)

(d) (e) (f)

Figure 3. Plots of numerical solution yh , ph , uh for for the discrete control variables (top) and plots of numerical solution yh , ph , uhfor variational discretization (bottom) withν = 0.01 h.

Moreover, we give the plots of exact and numerical solutions of twomethods at themesh h = 1/80in Figure 1 for the detail. From these figures, we can see that numerical solutions approximate theexact ones well.Case b.We consider the L-shaped domain in [0, 1] × [0, 1/2] ∪ [0, 1/2] × [1/2, 1]. The domain � isdivided by the triangulations of mesh size h = 1/16 in Figure 2(a). The coefficients β = (1, 2), α = 1and the admissible set is K = {u ∈ L2(�),−4 ≤ u ≤ 4}. The true solution is choosing the same one

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14 Z. WENG ET AL.

IsoValue0.09258180.2777450.4629090.6480730.8332361.01841.203561.388731.573891.759051.944222.129382.314542.499712.684872.870043.05523.240363.425533.61069

IsoValue0.04629090.1388730.2314540.3240360.4166180.50920.6017820.6943630.7869450.8795270.9721091.064691.157271.249851.342441.435021.52761.620181.712761.80534

IsoValue-9.75-9.25-8.75-8.25-7.75-7.25-6.75-6.25-5.75-5.25-4.75-4.25-3.75-3.25-2.75-2.25-1.75-1.25-0.75-0.25

IsoValue0.09260820.2778250.4630410.6482580.8334741.018691.203911.389121.574341.759561.944772.129992.315212.500422.685642.870863.056073.241293.42653.61172

IsoValue0.04614730.138760.2313730.3239860.4165990.5092120.6018240.6944370.787050.8796630.9722761.064891.15751.250111.342731.435341.527951.620571.713181.80579

IsoValue-9.74996-9.24988-8.7498-8.24972-7.74964-7.24956-6.74948-6.2494-5.74932-5.24924-4.74916-4.24909-3.74901-3.24893-2.74885-2.24877-1.74869-1.24861-0.748528-0.248449

IsoValue0.09260770.2778230.4630390.6482540.8334691.018681.20391.389121.574331.759551.944762.129982.315192.500412.685622.870843.056053.241273.426493.6117

IsoValue0.04630280.1389080.2315140.3241190.4167250.509330.6019360.6945410.7871470.8797520.9723581.064961.157571.250171.342781.435391.527991.62061.71321.80581

IsoValue-9.75-9.25-8.75-8.25-7.75-7.25-6.75-6.25-5.75-5.25-4.75-4.25-3.75-3.25-2.75-2.25-1.75-1.25-0.75-0.25

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4. Plots of numerical solution yh , ph , uh for the discrete control variables (top) and plots of numerical solution yh , ph , uh forvariational discretization (bottom) with nonlinear case.

as in the Case a. The convergence orders are shown by slopes in Figure 2(b) and (c), which validatethe theoretical results.Case c.Furthermore,we consider a problemwithout exact solutions. The parameters areβ = (1,

√3),

α = 0.1, and ε = 10−8. The admissible set isK = {u ∈ L2(�), 0.5 ≤ u ≤ 8}. The functions are f = 1and yd = 1. To examine the stable properties of the discrete scheme, Figure 3 shows the numericalsolutions by two methods on the mesh Th with h = 1/80.

Example 6.2: In this example, we consider the following nonlinear optimal control problem:

minu∈K12

∫�

(y − yd)2dx + α

2

∫�

u2dx, (53)

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APPLICABLE ANALYSIS 15

Table 3. Error estimates ν = 0.1 h for standard method.

1/h |||y − yh||| Rate |||p − ph||| Rate ‖u − uh‖0 Rate

20 7.97e−2 4.07e−2 9.77e−240 2.79e−2 1.51 1.42e−2 1.52 3.33e−2 1.5560 1.52e−2 1.51 7.67e−3 1.52 1.69e−2 1.6780 9.83e−3 1.51 4.96e−3 1.51 1.05e−2 1.65100 7.03e−3 1.50 3.54e−3 1.51 7.74e−3 1.37120 5.34e−3 1.50 2.69e−3 1.51 6.04e−3 1.36140 4.24e−3 1.50 2.13e−3 1.51 4.78e−3 1.52

Table 4. Error estimates ν = 0.1 h for standard method.

1/h |||y − yh||| Rate |||p − ph||| Rate ‖u − uh‖0 Rate

20 8.12e−2 4.04e−2 5.82e−240 2.82e−2 1.53 1.41e−2 1.52 1.46e−2 2.0060 1.52e−2 1.52 7.64e−3 1.51 6.48e−3 2.0080 9.87e−3 1.51 4.95e−3 1.51 3.65e−3 2.00100 7.05e−3 1.51 3.53e−3 1.51 2.33e−3 2.00120 5.38e−3 1.51 2.68e−3 1.51 1.62e−3 2.00140 4.25e−3 1.51 2.13e−3 1.51 1.19e−3 2.00

subject to:

β · ∇y − ε�y + y + y3 = f + u, in �, (54)y = 0, on ∂�. (55)

and the co-state elliptic equation as:

−β · ∇p − ε�p + 3y2p = y − yd , in �,p = 0, on ∂�.

Consider problem (53)–(55) with β = (2, 3), α = 0.1 and ε = 10−8. The computational domainis [0, 1] × [0, 1] and the admissible set is K = {u ∈ L2(�),−10 ≤ u ≤ 5}. The exact solutions aretaken as:

y = 100(1 − x1)x1(1 − x2)x22,p = 50(1 − x1)x1(1 − x2)x22,u = min{−10,max{5,−p/α}},

and f and yd are obtained by inserting y, p, u into the associated optimality system.In this example, the numerical solutions are computed on a series of triangular meshes, which are

created on uniformmesh. The errors are presented in Tables 3 and 4 for two methods, and two plotsof exact and numerical solutions of two methods are given at the mesh h = 1/80 in Figure 4.

7. Conclusions

In this paper, we discussed the bubble-stabilization Galerkin method for the constrained optimalcontrol problem governed by convection dominated diffusion equations. The main feature of ourstabilization method is using two local Gauss integrations to replace the projection operator, whichintroduces no additional variables. The discussion shows that our stabilization only depends on

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16 Z. WENG ET AL.

the bubble functions. Moreover, we obtain a priori error estimates of the standard method andthe variational discretization method. The numerical examples are presented to demonstrate ourtheoretical results. There are several possible extensions for this research. This method can beextended to the optimal control problem governed by nonlinear convection dominated diffusionequations or the Navier–Stokes equation. And the diffusion parameter independent analysis forsome special case deserves further investigations.

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which helped us to improvethe manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

The work of Z F Weng is partially supported by the Scientific Research Foundation of Huaqiao University [grantnumber 15BS307]. The work of J Z Yang is partially supported by National Natural Science Foundation of China[grant number 11171305], [grant number 91230203], and the research of X. Lu is partially supported by NationalNatural Science Foundation of China [grant number 91230108], [grant number 11471253].

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